Solve x^2-50x+336=0

$${ x }^{ 2 } -50x+336 = 0$$

Solve for x

$$x=8$$

$$x=42$$

Steps Using Factoring

To solve the equation, factor x^{2}-50x+336 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-50 ab=336$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 336.

$$-1,-336 -2,-168 -3,-112 -4,-84 -6,-56 -7,-48 -8,-42 -12,-28 -14,-24 -16,-21$$

Calculate the sum for each pair.

$$-1-336=-337 -2-168=-170 -3-112=-115 -4-84=-88 -6-56=-62 -7-48=-55 -8-42=-50 -12-28=-40 -14-24=-38 -16-21=-37$$

The solution is the pair that gives sum -50.

$$a=-42 b=-8$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-42\right)\left(x-8\right)$$

To find equation solutions, solve x-42=0 and x-8=0.

$$x=42 x=8$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+336. To find a and b, set up a system to be solved.

$$a+b=-50 ab=1\times 336=336$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 336.

$$-1,-336 -2,-168 -3,-112 -4,-84 -6,-56 -7,-48 -8,-42 -12,-28 -14,-24 -16,-21$$

Calculate the sum for each pair.

$$-1-336=-337 -2-168=-170 -3-112=-115 -4-84=-88 -6-56=-62 -7-48=-55 -8-42=-50 -12-28=-40 -14-24=-38 -16-21=-37$$

The solution is the pair that gives sum -50.

$$a=-42 b=-8$$

Rewrite x^{2}-50x+336 as \left(x^{2}-42x\right)+\left(-8x+336\right).

$$\left(x^{2}-42x\right)+\left(-8x+336\right)$$

Factor out x in the first and -8 in the second group.

$$x\left(x-42\right)-8\left(x-42\right)$$

Factor out common term x-42 by using distributive property.

$$\left(x-42\right)\left(x-8\right)$$

To find equation solutions, solve x-42=0 and x-8=0.

$$x=42 x=8$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-50x+336=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -50 for b, and 336 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 336}}{2}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 336}}{2}$$

Multiply -4 times 336.

$$x=\frac{-\left(-50\right)±\sqrt{2500-1344}}{2}$$

Add 2500 to -1344.

$$x=\frac{-\left(-50\right)±\sqrt{1156}}{2}$$

Take the square root of 1156.

$$x=\frac{-\left(-50\right)±34}{2}$$

The opposite of -50 is 50.

$$x=\frac{50±34}{2}$$

Now solve the equation x=\frac{50±34}{2} when ± is plus. Add 50 to 34.

$$x=\frac{84}{2}$$

Divide 84 by 2.

$$x=42$$

Now solve the equation x=\frac{50±34}{2} when ± is minus. Subtract 34 from 50.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

The equation is now solved.

$$x=42 x=8$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-50x+336=0$$

Subtract 336 from both sides of the equation.

$$x^{2}-50x+336-336=-336$$

Subtracting 336 from itself leaves 0.

$$x^{2}-50x=-336$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-336+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-336+625$$

Add -336 to 625.

$$x^{2}-50x+625=289$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=289$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{289}$$

Simplify.

$$x-25=17 x-25=-17$$

Add 25 to both sides of the equation.

$$x=42 x=8$$